Journal of Formalized Mathematics
Volume 8, 1996
University of Bialystok
Copyright (c) 1996 Association of Mizar Users

## Fixpoints in Complete Lattices

Piotr Rudnicki
University of Alberta, Edmonton
Andrzej Trybulec
Warsaw University, Bialystok

### Summary.

Theorem (5) states that if an iterate of a function has a unique fixpoint then it is also the fixpoint of the function. It has been included here in response to P. Andrews claim that such a proof in set theory takes thousands of lines when one starts with the axioms. While probably true, such a claim is misleading about the usefulness of proof-checking systems based on set theory.\par Next, we prove the existence of the least and the greatest fixpoints for $\subseteq$-monotone functions from a powerset to a powerset of a set. Scheme {\it Knaster} is the Knaster theorem about the existence of fixpoints, cf. . Theorem (11) is the Banach decomposition theorem which is then used to prove the Schr\"{o}der-Bernstein theorem (12) (we followed Paulson's development of these theorems in Isabelle ). It is interesting to note that the last theorem when stated in Mizar in terms of cardinals becomes trivial to prove as in the Mizar development of cardinals the $\leq$ relation is synonymous with $\subseteq$.\par Section 3 introduces the notion of the lattice of a lattice subset provided the subset has lubs and glbs.\par The main theorem of Section 4 is the Tarski theorem (43) that every monotone function $f$ over a complete lattice $L$ has a complete lattice of fixpoints. As the consequence of this theorem we get the existence of the least fixpoint equal to $f^\beta(\bot_L)$ for some ordinal $\beta$ with cardinality not bigger than the cardinality of the carrier of $L$, cf. , and analogously the existence of the greatest fixpoint equal to $f^\beta(\top_L)$.\par Section 5 connects the fixpoint properties of monotone functions over complete lattices with the fixpoints of $\subseteq$-monotone functions over the lattice of subsets of a set (Boolean lattice).

This work was partially supported by NSERC Grant OGP9207 and NATO CRG 951368.

#### MML Identifier: KNASTER

The terminology and notation used in this paper have been introduced in the following articles                     

#### Contents (PDF format)

1. Preliminaries
2. Fixpoints in general
3. The lattice of lattice subset
4. Complete lattices
5. Boolean lattices

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